This entry outlines the role of the law of non-contradiction (LNC) as
the foremost among the first (indemonstrable) principles of
Aristotelian philosophy and its heirs, and depicts the relation
between LNC and LEM (the law of excluded middle) in establishing the
nature of contradictory and contrary opposition. §1 presents the
classical treatment of LNC as an axiom in Aristotle's “First
Philosophy” and reviews the status of contradictory and contrary
opposition as schematized on the Square of Opposition. §2
explores in further detail the possible characterizations of LNC and
LEM, including the relevance of future contingent statements in which
LEM (but not LNC) is sometimes held to fail. In §3 I briefly
discuss the mismatch between the representation of contradictory
negation as a propositional operator and its varied realization within
natural language. §4 deals with several challenges to LNC within
Western philosophy, including the paradoxes, and the relation between
systems with truth-value gaps (violating LEM) and those with
truth-value gluts (violating LNC). Finally, in §5, the
tetralemma of Buddhist logic is discussed within the context of gaps
and gluts; it is argued that apparent violations of LNC in this
tradition and others can be attributed to either differing
viewpoints of evaluation (as foreseen by Aristotle) or to intervening
modal and epistemic operators.

The twin foundations of Aristotle's logic are the law of
non-contradiction (LNC) (also known as the law of contradiction, LC) and the
law of excluded middle (LEM). In Metaphysics Book Γ,
LNC—“the most certain of all principles”—is
defined as follows:

It is impossible that the same thing can at the same time
both belong and not belong to the same object and in the same respect,
and all other specifications that might be made, let them be added to
meet local objections (1005b19–23).

It will be noted that this statement of the LNC is an explicitly modal
claim about the incompatibility of opposed properties applying to the
same object (with the appropriate provisos). Since Łukasiewicz
(1910), this ontological version of the principle has been
recognized as distinct from, and for Aristotle arguably prior to,
the logical formulation (“The opinion that opposite
assertions are not simultaneously true is the firmest of
all”—Met. 1011b13–14) and the psychological
formulation (“It is impossible for anyone to believe that the
same thing is and is not, as some consider Heraclitus
said”—Met. 1005b23–25) offered elsewhere in Book Γ;
we return to Heraclitus below. Wedin (2004a), who argues for the
primacy of the ontological version (see also Meyer 2008, Other
Internet Resources), formalizes it as
¬◊(∃x)(Fx ∧
¬Fx). (See also
Aristotle on Non-contradiction.)

For Aristotle, the status of LNC as a first, indemonstrable principle
is obvious. Those who stubbornly demand a proof of LNC simply
“lack education”: since “a demonstration of
everything is impossible”, resulting in infinite regress. At
least some principles must be taken as primitive
axiomata rather than derived from other
propositions—and what principle more merits this status than
LNC? (1006a6–12). In first philosophy, as in mathematics, an axiom is
both indemonstrable and indispensable; without LNC, “a is
F” and “a is not F” are
indistinguishable and no argumentation is possible. While Sophists and
“even many physicists” may claim that it is
possible for the same thing to be and not to be at the same time and
in the same respect, such a position self-destructs “if only our
opponent says something”, since as soon as he opens his mouth to
make an assertion, any assertion, he must accept LNC. But
what if he does not open his mouth? Against such an
individual “it is ridiculous to seek an argument” for he
is no more than a vegetable (1006a1–15).

The celebrated Arab commentator Avicenna (ibn Sīnā,
980–1037) confronts the LNC skeptic with a more severe fate than
Aristotle's vegetable reduction: “As for the obstinate, he must
be plunged into fire, since fire and non-fire are identical. Let him
be beaten, since suffering and not suffering are the same. Let him be
deprived of food and drink, since eating and drinking are identical to
abstaining” (Metaphysics I.8, 53.13–15).

The role of LNC as the basic, indemonstrable “first
principle” is affirmed by Leibniz, for whom LNC is taken as
interdefinable with the Law of Identity that states that everything is
identical to itself: “Nothing should be taken as first
principles but experiences and the axiom of identity or (what is the
same thing) contradiction, which is primitive, since otherwise there
would be no difference between truth and falsehood, and all
investigation would cease at once, if to say yes or no were a matter
of indifference” (Leibniz 1696/Langley 1916: 13–14). For
Leibniz, everybody—even “barbarians”—must
tacitly assume LNC as part of innate knowledge implicitly called upon
at every moment, thus demonstrating the insufficiency of Locke's
empiricism (ibid.,
77).[1]

In accounting for the incompatibility of truth and falsity, LNC lies
at the heart of the theory of opposition, governing both
contradictories and contraries.
(See traditional square of opposition.)
Contradictory opposites (“She is sitting”/“She is
not sitting”) are mutually exhaustive as well as mutually
inconsistent; one member of the pair must be true and the other
false. As it was put by the medievals, contradictory opposites divide
the true and the false between them; for Aristotle, this is the
primary form of
opposition.[2]
Contrary opposites (“He is happy”/“He is
sad”) are mutually inconsistent but not necessarily exhaustive;
they may be simultaneously false, though not simultaneously true. LNC
applies to both forms of opposition in that neither contradictories
nor contraries may belong to the same object at the same time and in
the same respect (Metaphysics 1011b17–19). What distinguishes
the two forms of opposition is a second indemonstrable principle, the
law of excluded middle (LEM): “Of any one subject, one thing
must be either asserted or denied” (Metaphysics
1011b24). Both laws pertain to contradictories, as in a paired
affirmation (“S is P”) and denial
(“S isn't P”): the negation is true
whenever the affirmation is false, and the affirmation is true when
the negation is false. Thus, a corresponding affirmation and negation
cannot both be true, by LNC, but neither can they both be
false, by LEM. But while LNC applies both to contradictory
and contrary oppositions, LEM holds only for contradictories:
“Nothing can exist between two contradictories, but something
may exist between contraries” (Metaphysics 1055b2): a
dog cannot be both black and white, but it may be neither.

As Aristotle explains in the Categories, the opposition
between contradictories—“statements opposed to each other
as affirmation and negation”—is defined in two
ways. First, unlike contrariety, contradiction is restricted to
statements or propositions; terms are never related as
contradictories. Second, “in this case, and in this case only,
it is necessary for the one to be true and the other false”
(13b2–3).

Opposition between terms cannot be contradictory in
nature, both because only statements (subject-predicate combinations)
can be true or false (Categories 13b3–12) and because any two
terms may simultaneously fail to apply to a given
subject.[3]
But two statements may be members of either a contradictory or a
contrary opposition. Such statements may be simultaneously false,
although (as with contradictories) they may not be simultaneously
true. The most striking aspect of the exposition for a modern reader
lies in Aristotle's selection of illustrative material. Rather than
choosing an uncontroversial example involving mediate
contraries, those allowing an unexcluded middle (e.g. “This dog
is white”/“This dog is black”; “Socrates is
good”/“Socrates is bad”), Aristotle offers a pair of
sentences containing immediate contraries, “Socrates is
sick”/“Socrates is well”. These propositions may
both be false, even though every person is either ill or well:
“For if Socrates exists, one will be true and the other false,
but if he does not exist, both will be false; for neither
‘Socrates is sick’ nor ‘Socrates is well’ will
be true, if Socrates does not exist at all” (13b17–19). But
given a corresponding affirmation and negation, one will always be
true and the other false; the negation “Socrates is not
sick” is true whether the snub-nosed philosopher is healthy or
non-existent: “for if he does not exist, ‘he is
sick’ is false but ‘he is not sick’ true”
(13b26–35).

Members of a canonical pair of contradictories are formally identical
except for the negative particle:

An affirmation is a statement affirming something of something, a
negation is a statement denying something of something…It is
clear that for every affirmation there is an opposite negation, and
for every negation there is an opposite affirmation…Let us call
an affirmation and a negation which are opposite a
contradiction (De Interpretatione 17a25–35).

But this criterion, satisfied simply enough in the case of singular
expressions, must be recast in the case of quantified expressions,
both those which “signify universally” (“every
cat”, “no cat”) and those which do not (“some
cat”, “not every cat”).

For such cases, Aristotle shifts from a formal to a semantically based
criterion of opposition (17b16–25). Members of
an A/O pair (“Every man is
white”/“Not every man is white”)
or I/E pair (“Some man is
white”/“No man is white”) are contradictories
because in any state of affairs one member of each pair must be true
and the other false. Members of an A/E
pair—“Every man is just”/“No man is
just”—constitute contraries, since these cannot be true
but can be false together. The contradictories of two contraries
(“Not every man is just”/“Some man is just”)
can be simultaneously true with reference to the same subject;
(17b23–25). This last opposition of I and O
statements, later to be dubbed subcontraries because they
appear below the contraries on the traditional square, is a peculiar
opposition indeed; Aristotle elsewhere (Prior Analytics
63b21–30) sees I and O as “only verbally
opposed”, given the mutual consistency of e.g. “Some
Greeks are bald” and “Some Greeks aren't bald” (or
“Not all Greeks are bald”, which doesn't necessarily
amount to the same thing, given existential import;
see traditional square of opposition).

The same relations obtain for modal propositions, those involving
binary connectives like “and” and “or”,
quantificational adverbs, and a range of other expressions that can be
mapped onto the square in analogous ways (see Horn 1989). Thus for example we have the
following modal square, based on De Interpretatione
21b10ff. and Prior Analytics 32a18–28:

(1) Modal Square

In the twelfth century, Peter of Spain (1972: 7) offers a
particularly elegant formulation in his Tractatus; it will be
seen that these apply to the modal propositions in (1) as well as to
the quantificational statements in the original square:

The law of contradictories is such that if one
contradictory is true the other is false and vice versa, for nothing
can be simultaneously true and false.

Each contradictory is equivalent to the
negation of the other.

Each contradictory entails and is entailed by
the negation of the other.

The law of contraries is such that if one is true the other is false
but not vice versa.

Each contrary statement entails the negation of the other but
not vice versa. [E.g. “I am happy” unilaterally entails
“I am not unhappy”; “It is necessary that
Φ” unilaterally entails “It is not impossible that
Φ”.]

The law of subcontraries is such that if one is false the other is
true but not vice versa.

The law of excluded middle, LEM, is another of Aristotle's first
principles, if perhaps not as first a principle as
LNC. Just as Heraclitus's anti-LNC position, “that everything is
and is not, seems to make everything true”, so too Anaxagoras's
anti-LEM stance, “that an intermediate exists between two
contradictories, makes everything false” (Metaphysics
1012a25–29). Of any two contradictories, LEM requires that one must be
true and the other false (De Interpretatione 18a31)—or
does it? In a passage that has launched a thousand treatises,
Aristotle (De Interpretatione, Chapter 9) addresses the
difficulties posed by apparently contradictory contingent statements
about future events, e.g. (2a,b).

(2a) There will be a sea-battle tomorrow.
(2b) There will not be a sea-battle tomorrow.

Clearly, (2a) and (2b) cannot both be true; LNC applies to future
contingents as straightforwardly as to any other pair of
contradictories. But what of LEM? Here is where the difficulties
begin, culminating in the passage with which Aristotle concludes and
(apparently) summarizes his account:

It is necessary for there to be or not to be a sea-battle
tomorrow; but it not necessary for a sea-battle to take place
tomorrow, nor for one not to take place—though it is necessary
for one to take place or not to take place. So, since statements are
true according to how the actual things are, it is clear that wherever
these are such as to allow of contraries as chance has it, the same
necessarily holds for the contradictories also. This happens with
things that are not always so or are not always not so. With these it
is necessary for one or the other of the contradictories to be true or
false—not, however, this one or that one, but as chance has it;
or for one to be true rather than the other, yet not
already true or false. Clearly, then it is not necessary
that of every affirmation and opposite negation one should be true and
the other false. For what holds for things that are does not hold for
things that are not but may possibly be or not be; with these it is as
we have said. (De Interpretatione 19a30-b4)

Unfortunately, given the systematic ambiguity and textual variations
in the Greek text, the difficulty of telling when Aristotle is
speaking with his own voice or characterizing an opponent's argument,
and the lack of formal devices for the essential scopal distinctions
at issue, it has never been clear exactly just what has been said here
and in the chapter more generally. Some, including Boethius and
Lukasiewicz, have seen in this text an argument for rejecting LEM for
future contingent statements, which are therefore to be assigned a
non-classical value (e.g. “Indeterminate”) or no
truth-value at
all.[4]
Their reasoning is based in part on the premise that the alternative
position seems to require the acceptance of determinism. Others,
however, read Aristotle as rejecting not simple bivalence for future
contingents but rather determinacy itself. This interpretive
tradition, endorsed by al-Fārābi, Saint Thomas, and
Ockham, is crystallized in this passage from Abelard's
Dialectica (210–22) cited by Kneale and Kneale (1962:
214):

No proposition de contingenti futuro can be determinately
true or determinately false…, but this is not to say that
no such proposition can be true or false. On the contrary, any such
proposition is true if the outcome is to be true as it states, even
though this is unknown to us.

Even if we accept the view that Aristotle is uncomfortable with
assigning truth (or falsity) to (2a) and (2b), their disjunction
in (3a) is clearly seen as true, and indeed as necessarily true.
But the modal operator must be taken to apply to the disjunction as a
whole as in (3b) and not to each disjunct as in (3c).

(3a) Either there will be or there will not be a sea-battle tomorrow.
(3b) □ (Φ
∨
¬Φ)
(3c) □ Φ
∨
□ ¬Φ

For Aristotle, LNC is understood primarily not as the principle of propositional
logic that no statement can be true simultaneously with its negation,
but as a prima facie rejection of the possibility that any predicate
F could both hold and not hold of a given subject (at the
same time, and in the same respect). A full rendering of the version
of LNC appearing at Metaphysics 1006b33–34—“It is
not possible to truly say at the same time of a thing that it is a man
and that it is not a man”—would require a representation
involving operators for modality and truth and allowing quantification
over
times.[5]
In the same way, LEM is not actually the principle that every
statement is either true or has a true negation, but the law that for
any predicate F and any entity x, x either
is F or isn't F.

But these conceptualizations of LNC and LEM must be generalized, since
the principle that it is impossible for a to be F and not to
be F will not apply to statements of arbitrary complexity. We
can translate the Aristotelian language, with some loss of
faithfulness, into the standard modern propositional versions in (4a,b)
respectively, ignoring the understood modal and temporal
modifications:

(4a) LNC: ¬(Φ ∧ ¬Φ)
(4b) LEM: Φ
∨
¬Φ

Taking LNC and LEM together, we obtain the result that exactly one
proposition of the pair {Φ, ¬Φ} is true and exactly one is
false, where ¬ represents contradictory negation.

Alternatively, the laws can be recast semantically as in (5), again
setting aside the usual qualifications:

(5a) LNC: No proposition may be simultaneously true and false.
(5b) LEM: Every proposition must be either true or false.

Not every natural language negation is a contradictory operator, or
even a logical operator. A statement may be rejected as false, as
unwarranted, or as inappropriate—misleading, badly pronounced,
wrongly focused, likely to induce unwanted implicatures or
presuppositions, overly or insufficiently formal in register. Only in
the first of these cases, as a toggle between truth and falsity, is it
clear that contradictory negation is involved (Horn 1989, Smiley
1993). Sainsbury (2004) takes truth-functional contradictory negation
to be a special case of a generalized option negation as a
deselection operator: If there are two mutually exhaustive and
exclusive options A and B, to select A is to deselect B. But the
relevant options may involve not truth, but some other aspect of
utterance form or meaning as in the standard examples
of metalinguistic negation (Horn 1989): “That's not a
car, it's a Volkswagen”, “Cancer selection is not a but
the major force in the emergence of complex animal life”,
“He's not your old man, he's your father”, “We
didn't call the POlice, we called the poLICE”. In such cases,
the relevant target for deselection is what the right thing is to say
in a particular context, where “truth is not sufficient for
being right, and may not even be necessary” (Sainsbury 2004:
87). Thus the apparent LNC violation (if it's a Volkswagen, it both is
and isn't a car) is not a real one.

Given that not every apparent sentential negation is contradictory, is
every contradictory negation sentential? Within propositional logic,
contradictory negation is a self-annihilating operator:
¬(¬Φ) is equivalent to Φ. This is explicitly
recognized in the proto-Fregean Stoic logic of Alexander of
Aphrodisias: “‘Not: not: it is day’ differs from
‘it is day’ only in manner of speech” (Mates 1953:
126). The Stoics' apophatikon directly prefigures the
iterating and self-cancelling propositional negation of Frege and
Russell. As Frege puts it (1919: 130), “Wrapping up a thought in
double negation does not alter its truth value.” The
corresponding linguistic principle is expressed in the grammarians'
bromide, “Duplex negatio affirmat.”

Not all systems of propositional logic accept a biconditional law of
double negation (LDN), ¬(¬Φ) ≡ Φ. In particular,
LDN, along with LEM, is not valid for the Intuitionists, who reject
¬(¬Φ) → Φ while accepting its converse, Φ
→ ¬(¬Φ). But the very possibility of applying
negation to a negated statement presupposes the analysis of
contradictory negation as an iterative operator (one capable of
applying to its own output), or as a function whose domain is
identical to its range. Within the categorical term-based logic of
Aristotle and his Peripatetic successors, every
statement—whether singular or general—is of
subject-predicate form. Contradictory negation is not a one-place
operator taking propositions into propositions, but rather a mode
of predication, a way of combining subjects with predicates: a
given predicate can be either affirmed or denied of a given
subject. Unlike the apophatikon or propositional negation
connective introduced by the Stoics and formalized in Fregean and
Russellian logic, Aristotelian predicate denial, while toggling truth
and falsity and yielding the semantics of contradictory opposition,
does not apply to its own output and hence does not syntactically
iterate. In this respect, predicate denial both anticipates the form
of negation in Montague Grammar and provides a more plausible
representation of contradictory negation in natural language, whether
Ancient Greek or English, where reflexes of the iterating one-place
connective of the Stoics and Fregeans (“Not: not: the sun is
shining”) are hard to find outside of artificial constructs like
the “it is not the case” construction (Horn 1989,
§7.2). In a given natural language, contradictory negation may be
expressed as a particle associated with a copula or a verb, as an
inflected auxiliary verb, as a verb of negation, or as a negative
suffix or prefix.

In addition, there is a widespread pragmatically motivated tendency
for a formal contradictory negation to be strengthened to a semantic
or virtual contrary through such processes as litotes (“I don't
like prunes” conveying that I dislike prunes) and so-called
neg(ative) raising (“I don't think that Φ” conveying
“I think that ¬Φ”). Similarly, the prefixal
negation in such adjectives as “unhappy” or
“unfair” is understood as a contrary rather than
contradictory (not-Adj) of its base. These phenomena have been much
discussed by rhetoricians, logicians, and linguists (see Horn 1989:
Chap. 5).

In addition to predicate denial, in which a predicate F is
denied of a subject a, Aristotelian logic allows for
narrow-scope predicate term negation, in which a negative
predicate not-F is affirmed of a. The relations of
predicate denial and predicate term negation to a simple affirmative
proposition (and to each other) can be schematized on a generalized
square of opposition for singular (non-quantified) expressions (De
Interpretatione 19b19–30, Prior Analytics Chapter
46):

(6) Negation Square

If Socrates doesn't exist, “Socrates is wise” (A)
and its contrary “Socrates is not-wise” (E) are
both automatically false (since nothing—positive or
negative—can be truly affirmed of a non-existent subject), while
their respective contradictories “Socrates is not wise”
(O) and “Socrates is not not-wise” (I)
are both true. Similarly, for any object x, either x
is red or x is not red—but x may be neither
red nor not-red; if, for instance, x is a unicorn or a prime
number.

While Russell (1905) echoed (without acknowledgment) Aristotle's
ambiguist analysis of negation as either contradictory
(“external”) or contrary (“internal”), by
virtue of the two logical forms assigned to “The king of France
is not bald” (see descriptions),
such propositionalized accounts are bought at a cost of naturalness,
as singular sentences of subject-predicate grammatical form are
assigned the logical form of an existentially quantified conjunction
and as names are transmuted into predicates.

While Aristotle would see a republican France as rendering (7a) false
and (7b) automatically true, Frege (1892) and Strawson (1950) reject
the notion that either of these sentences can be used to make a true
or false assertion. Instead, both statements presuppose the existence
of a referent for the singular term; if the presupposition fails, so
does the possibility of classical truth assignment. Note, however,
that such analyses present a challenge to LEM only if (7b) is taken as
the true contradictory of (7a), an assumption not universally
shared. Russell, for example, allows for one reading of (7b) on which
it is, like (7a), false in the absence of a referent or denotatum for
the subject term; on that reading, on which the description has
primary occurrence, the two sentences are not contradictories. In this
way, Russell (1905: 485) seeks to guide the French monarch out of the
apparent trap without recourse to wigs or truth value gaps:

By the law of the excluded middle, either ‘A is B’ or
‘A is not B’ must be true. Hence either ‘the present
king of France is bald’ or ‘the present king of France is
not bald’ must be true. Yet if we enumerated the things that are
bald and the things that are not bald, we should not find the king of
France on either list. Hegelians, who love a synthesis, will probably
conclude that he wears a wig.

In those systems that do embrace truth value gaps (Strawson, arguably
Frege) or non-classically-valued systems (Łukasiewicz, Bochvar,
Kleene), some sentences or statements are not assigned a (classical)
truth value; in Strawson's famous dictum, the question of the truth
value of “The king of France is wise”, in a world in which
France is a republic, simply fails to arise. The negative form of such
vacuous statements, e.g. “The king of France is not wise”,
is similarly neither true nor false. This amounts to a rejection of
LEM, as noted by Russell 1905. In addition to vacuous singular
expressions, gap-based analyses have been proposed for future
contingents (following one reading of Aristotle's exposition of the
sea-battle; cf. §2 above) and category mistakes (e.g. “The
number 7 likes/doesn't like to dance”).

While LNC has traditionally remained more sacrosanct, reflecting its
position as the primus inter pares of the indemonstrables,
transgressing this final taboo has become increasingly alluring in
recent years. The move here involves embracing not gaps but truth
value gluts, cases in which a given sentence and its negation
are taken to be both true, or alternatively cases in which a sentence
may be assigned more than one (classical) truth value, i.e. both True
and False. Parsons (1990) observes that the two non-classical theories
are provably logically equivalent, as gluts arise within one class of
theories precisely where gaps do in the other; others, however, have
argued that gaps (as in Intuitionist non-bivalent logics) are easier
to swallow than gluts (see papers in the Priest et al. 2004 collection
for further debate). Dialetheists reject the charge of incoherence by
noting that to accept some contradictions is not to accept them all;
in particular, they seek to defuse the threat of logical armageddon or
“explosion” posed by Ex Contradictione Quodlibet, the
inference in (8):

(8) p, ¬p
_____
∴q

Far from reduced to the silence of a vegetable, as Aristotle ordained,
the proponents of true contradictions, including self-avowed
dialetheists following the lead of Sylvan (né Routley) and
Priest have been eloquent.

Is the status of Aristotle's “first principle” as obvious
as he believed? Adherents of the dialetheist view that there are true
contradictories (Priest 1987, 1998, 2002; see also
dialetheism and
paraconsistent logic)
would answer firmly in the
negative.[6]
In the Western tradition, the countenancing of true contradictions is
typically—although not exclusively—motivated on the basis
of such classic logical paradoxes as “This sentence is not
true” and its analogues (the Liar, the Barber, Russell's
paradox), each of which is true if and only if it is not true. As
Smiley (1993: 19) has remarked, “Dialetheism stands to the
classical idea of negation like special relativity to Newtonian
mechanics: they agree in the familiar areas but diverge at the margins
(notably the paradoxes).”

Related to the classic paradoxes of logic and set theory is the
Paradox of the Stone. One begins by granting the basic dilemma, as an
evident instance of LEM: either God is omnipotent or God is not
omnipotent. With omnipotence, He can do anything, and in particular He
can create a stone, call it s, that is so heavy even He
cannot lift it. But then there is something He cannot do, viz. (ex
hypothesi) lift s. But this is a violation of LNC: God can
lift s and God cannot lift s. This paradox, and the
potential challenge it offers to either LNC or the possibility of
omnipotence, has been recognized since Aquinas, who opted for
retaining the Aristotelian law by understanding omnipotence as the
capacity to do only what is not logically impossible. (Others,
including Augustine and Maimonides, have noted that in any case God is
“unable” to do what is inconsistent with His nature,
e.g. commit sin.) For Descartes, on the other hand, an omnipotent God
is by definition capable of any task, even those yielding
contradictions. Mavrodes (1963), Kenny, and others have sided with
St. Thomas in taking omnipotence to extend only to those powers it is
possible to possess; Frankfurt (1964), on the other hand, essentially
adopts the Cartesian line: Yes, of course God can indeed construct a
stone such that He cannot lift it—and what's more, He can lift
it! (See also Savage 1967 for a related solution.)

As we have seen, the target of Aristotle's psychological (doxastic)
version of LNC was Heraclitus: “It is impossible for anyone to
believe that the same thing is and is not, as some consider Heraclitus
said—for it is not necessary that the things one says one also
believes” (Met. 1005b23–26). But as Aristotle acknowledges here
(even if he less politic elsewhere), there is considerable uncertainty
over exactly what Heraclitus said and what he believed. Heraclitus
could not have literally rejected LNC, as he is often accused of (or
praised for) doing, as his writings preceded the statement of that
principle in Metaphysics Γ by well over a century. But
the question remains: do his words, as represented in the extant
fragments, anticipate the Dialetheists and other rejectionists? Yes
and no. To be sure, Heraclitus was proud to wear the mantle of
“paradoxographer” (Barnes 1982: 80) and enjoyed nothing
more than to épater les bourgeois of his day. But the
key fragments supporting his proclamation of the Unity of Opposites
can be taken in more than one way. He points out that sea water is
salutary (if you're a fish) and unhealthy (if you're a human), just as
garbage is preferable to gold (for a donkey) but then again it isn't
(for a person). And given the inevitability of flux (as Heraclitus
memorably illustrated by his river into which one cannot step twice),
what is true (today) is also false (tomorrow). But these astute
observations do not so much refute the LNC as much as demonstrate
(through what Barnes calls the Fallacy of the Dropped Qualification)
the need for Aristotle's crucial codicil: sea water, for example,
cannot be both healthful and unhealthful for the same experiencer at
the same time and in the same respect. (In a similar way, “It is
raining” may of course be deemed true at this moment in Seattle
and false in Palo Alto; we need admit no contradiction here, whether
we deal with the issue by endorsing unarticulated constituents or in
some other manner; cf. Recanati 2002 inter alia.) Ultimately, whether
one follows Kirk (Heraclitus [1954]) in charging Aristotle with
misrepresenting Heraclitus as an LNC-denier or sides with Barnes
(1982) and Wedin (2004b) in sustaining Aristotle's accusation, it is
hard to see in what respect the evidence presented by Heraclitus,
however subtle a guide he may be for our travels on that path on which
up and down is one and the same, threatens the viability of LNC. (See
also Heraclitus.)

Within the modern philosophical canon, Hegel has often been seen as
the echt LNC-skeptic, well before his reputed deathbed lament,
“Only one man ever understood me, and he didn't understand
me.” Hegel saw himself as picking up where Heraclitus left
off—“There is no proposition of Heraclitus which I have
not adopted in my logic” (Barnes 1982: 57)—and indeed the
Heraclitean view of a world shaped by the unity of opposites through
strife and resolution does seem to foreshadow Hegelian dialectic. In
fact, however, an unresolved contradiction was a sign of error for
Hegel. The contradiction between thesis and antithesis results in the
dialectical resolution or superseding of the contradiction between
opposites as a higher-level synthesis through the process
of Aufhebung (from aufheben, a verb simultaneously
interpretable as 'preserve, cancel, lift up'). Rather than repudiating
LNC, Hegel's dialectic rests upon it. In Marxist theory, too,
contradictories do not simply cancel out but are dynamically resolved
(aufgehoben) at a higher level in a way that both preserves
and supersedes the contradiction, motivating the historical
dialectic. (See Horn 1989: §1.3.2.)

For Freud, there is a realm in which LNC is not so much superseded but
dissolved. On the primary, infantile level, reflected in dreams and
neuroses, there is no not: “‘No’ seems not
to exist as far as dreams are concerned. Anything in a dream can mean
its contrary” (Freud 1910: 155). When the analysand insists of a
dream character “It's not my mother”, the analyst
knowingly translates, “So it is his mother!”
Freud sought to ground this pre-logical, LNC-free (and negation-free)
realm not just in the primal realm of the dreamer's unconscious but
also in the phenomenon of Gegensinn, words
(especially Urworte, primal words) with two opposed meanings
purported attested widely in ancient and modern languages. The
empirical basis for this latter claim, however, has been widely
discredited; see Benveniste 1956.

Given Aristotle's observation (Metaphysics1006a2)that
“even some physicists” deny LNC and affirm that is indeed
possible for the same thing to be and not to be at the same time and
in the same respect, he would not have been surprised to learn that
quantum mechanics has made such challenges fashionable again. Thus, we
have Schrödinger's celebrated imaginary cat, placed (within the
context of a thought experiment) inside a sealed box along with
radioactive material and a vial of poison gas that will be released if
that material decays. Given quantum uncertainty, an atom inhabits both
states—decayed and non-—simultaneously, rendering the cat
(in the absence of an observer outside the system) both alive and
dead. Where speculative consensus breaks down is on whether
Schrödinger's paradox arises only when the quantum system is
isolated from the environment.

As we have seen, Aristotle himself anticipated many of the challenges
that have since been raised against LNC. One more such challenge is
posed by the ubiquity of doxastic inconsistency. Take the desires of
Oedipus, for example. In seeking Jocasta as his mate, did he wish to
marry his mother? Certainly he did on the de re reading: Oedipus's
mother (Jocasta) is such that he wanted to marry her, although he
would not have assented to the claim that he wanted to marry his
mother. In a sense, then, “Oedipus wanted to marry his
mother” is true (de re) and false (de dicto), but no violation
of LNC is incurred, since these represent different propositions, the
semantic distinction neutralized within the sentential form. But what
of the de dicto reading itself: it is really false? After
all, as a young boy Oedipus can be assumed (by some) to have exhibited
the eponymous complex, according to which the falsity of the (de
dicto) proposition that he wanted to marry his mother on
a conscious level belies the truth of this proposition on
an unconscious level. But this does not entail that he both
wished and did not wish to “marry” his mother at the same
time and in the same respect. Whether it concerns the
unacknowledged incestuous conflict of the Theban king, the indecision
of Zerlina's “Vorrei e non vorrei” response to Don
Giovanni's invitation, or the unspecified ambivalence of the
respondent in Strawson's exchange (1952: 7)

—Were you pleased?
—Well, I was and I wasn't.

we have ample opportunity to reflect
on the foresight of Aristotle's rider: “a is
F” and “a is not F”
cannot both hold in the same sense, at the same time, and in the
same respect.

Beyond the Western canon, the brunt of the battle over LNC has been
largely borne by the Buddhists, particularly in the exposition by
Nāgārjuna of the catuṣkoṭi or tetralemma (c. 200
A.D.; cf. Bochenski 1961: Part VI, Raju 1954, Garfield 1995, Tillemans
1999, Garfield & Priest 2002), also known as the four-cornered or
fourfold negation. Consider the following four possible truth
outcomes for any statement and its (apparent) contradictory:

(9) (i)

S is P

(ii)

S is not P

(iii)

S is both P and not-P

(iv)

S is neither P nor not-P

For instances of the positive tetralemma, on Nāgārjuna's
account, all four statement types can or must be accepted:

Everything is real and not real.
Both real and not real.
Neither real nor not real.
That is Lord Buddha's teaching.
—Mūla-madhyamaka-kārikā
18:8, quoted in Garfield (1995: 102)

Such cases arise only when we are beyond the realm to which ordinary
logic applies, when “the sphere of thought has ceased.” On
the other hand, much more use is made of the negative tetralemma, in
which all four of the statements in (9) can or must be rejected. Is
this tantamount, as it appears, to the renunciation of LEM and LNC, the
countenancing of both gaps and gluts, and thus—in
Aristotle's view—the overthrow of all bounds of rational
argument?

It should first be noted that the axiomatic status of LNC and LEM is
as well-established within the logical traditions of India as it is
for the Greeks and their
epigones.[7]
Garfield (1995) and Tillemans (1999) convincingly refute
the claim that Nāgārjuna was simply an
“irrationalist”.[8]
In the first place, if Nāgārjuna simply rejected LNC, there
would be no possibility of reductio arguments, which hinge on
the establishment of untenable contradictions, yet such arguments are
standardly employed in his logic. In fact, he explicitly
prohibits virodha (contradiction). Crucially, it is only in
the realm of the Absolute or Transcendent, where we are contemplating
the nature of the ultimate, that contradictions are embraced; in the
realm of ordinary reality, LNC operates and classical logic
holds. (Recall Freud's dichotomy between the LNC-observant conscious
mind and the LNC-free unconscious.) In this sense, the logic of
Nāgārjuna and of the Buddhist tradition more generally can
be seen not as inconsistent but paraconsistent. Indeed, just as
Aristotle ridiculed LNC-skeptical sophists as no better than
vegetables (see §1), the Buddhists dismissed the arch-skeptic
Sanjaya and his followers, who refused to commit themselves to a
definite position on any issue, as
“eel-wrigglers” (amarāvikkhepa). Sanjaya
himself was notorious for his periodic lapses into the extended
silence Aristotle described as the last refuge of the LNC-skeptic (see
Raju 1954).

One aspect of the apparent paradox is precisely parallel to that
arising with some of the potential counterexamples to the LNC arising
in Western thought. In various Buddhist and Jainist systems of
thought, the apparent endorsement of Fa &
¬Fa (or, in propositional terms, Φ ∧ ¬Φ)
is upon closer examination qualified in precisely the way foreseen by
the codicils in Aristotle's statement of the law: From a certain
viewpoint, Φ (e.g. Nirvana exists); from a certain viewpoint,
¬Φ(e.g. Nirvana does not exist). (Compare the observation of
Jainists two millennia ago that “S is P”
and “S is not
P” can both be true from different standpoints; cf. Raju
1954: 698–701; Balcerowicz 2003.)

To further explore the status of truth-value gluts, in which both
classical values are simultaneously assigned to a given proposition
(e.g. “x is real”), let us consider the analogous
cases involving gaps. Recall, for example, the case of future
contingents as in (2a,b) above: we need not maintain that “Iraq
will become a secular democracy” is neither true nor false when
uttered today, but only that neither this statement nor its
contradictory “Iraq will not become a secular democracy” is
assertable today in the absence of foreknowledge. Similarly
for past unknowables, such as (to adapt an example from Quine) the
proposition that the number of blades of grass on the Old Campus lawn
during the 2005 Yale commencement exercises was odd. This is again
more plausibly viewed as unassertable than as truth-valueless, even
though its truth-value will never be known. To take a third example,
we can argue, with Grice (1989: 80ff.), that a negation outside the
scope of a conditional is generally intended as a refusal (or
hesitation) to assert “if p then q”
rather than as the contradictory negation of a conditional, whose
truth value is determined in accord with the standard material
equivalence:

(10) ¬(p→q) ≡ (p &
¬q)

Thus, in denying your conditional “If you give her penicillin,
she will get better”, I am allowing for the possibility that
giving her penicillin might have no effect on her, but I am not
predicting that you will administer the penicillin and she will fail
to recover. Nor does denying the apothegm (typically though
inaccurately attributed to Dostoyevsky or Nietzsche) that if God is
dead everything is permitted commit one to the conjoined proposition
that God is dead and something is forbidden. As Dummett (1973: 328–30)
puts the point, we must distinguish negation outside the scope of a
Fregean assertion operator, not
(⊢p),
from the assertion of a negative proposition,
⊢(not p).
The former interpretation “might be taken to be a means of
expressing an unwillingness to assert” p, in particular
when p is a conditional:

(11) X:

If it rains, the match will be canceled.

Y:

That's not so. (or, I don't think that's the case.)

Y's contribution here does not constitute a negation of
X's content; rather, we can paraphrase Y as
conveying (11′a) or (11′b):

(11′a) If it rains, the match won't necessarily be canceled.
(11′b) It may [epistemic] happen that it rains and yet
the match is not canceled.

Dummett observes, “We have no negation of the conditional of
natural language, that is, no negation of its sense: we have only a
form for expressing refusal to assent to its assertion.”

Similarly with disjunction. Consider the exchange in (12) preceding
the 2000 election, updated from an example of Grice:

(12) X:

Bush or Gore will be elected.

Y:

That's not so: Bush or Gore or Nader will be elected.

Y's rejoinder cannot be a contradictory of the content of
X's claim, since the (de jure) election of Bush rendered both
X's and Y's statements true. Rather, Y
objects on the grounds that X is not in an epistemic position
to assert the binary disjunction.

Unassertability can be read as the key to the apparent paradox of the
catuṣkoṭi as well. The venerable text in
Majjhima-nikāya 72, relating the teachings of the
historical Buddha, offers a precursor for Nāgārjuna's
doctrine of the negative tetralemma. Gotama is responding to a monk's
question concerning the doctrine of rebirth (quoted in Robinson 1967:
54):

Gotama, where is the monk reborn whose mind is thus freed?
Vaccha, it is not true to say that he is
reborn.
Then, Gotama, he is not reborn.
Vaccha, it is not true to say that he is not
reborn.
Then, Gotama, he is both reborn and not reborn.
Vaccha, it is not true to say that he is both
reborn and not reborn.
Then, Gotama, he is neither reborn nor not reborn.
Vaccha, it is not true to say that he is neither
reborn nor not reborn.

Note the form of the translation here, or similarly that of the
standard rendering of the negative catuṣkoṭi that “it
profits not” to assert Φ, to assert ¬Φ, to assert
both Φ and ¬Φ, or to assert neither Φ nor ¬Φ:
the relevant negation can be taken to operate over an implicit modal,
in particular an epistemic or assertability operator. If so, neither
LEM nor LNC is directly at stake in the tetralemma: you can have your
Aristotle and Buddha too.

We tend to recalibrate apparent violations of LNC as conforming to a
version of the law that incorporates the Aristotelian qualifications:
a sincere defense of “p and not-p”
plausibly involves a change in the context of evaluation or a shift in
viewpoint, or alternatively a suppression of modal or epistemic
operators. This practice can be seen as an instance of a general
methodological principle associated with
Davidson
and Quine that has come to be called the principle of
charity (or, alternately, the principle of rational
accommodation): when it is unclear how to interpret another's
argument, interpret it in a way that makes the most sense. At the same
time, this procedure evokes the standard Gricean mode of explanation:
granted the operation of the Cooperative Principle and, more broadly,
the shared premise of rationality, we reinterpret apparent violations of
valid principles or maxims so as to conserve the assumption that one's
interlocutor is a rational and cooperative agent. And as Aristotle would
remind us, no principle is more worthy of conservation than the Law of
Non-Contradiction.